Problem 9 - Algebra 1: Concepts and Skills

Question

Find and correct the error. $$\frac{x+3}{x-3} \div \frac{4 x}{x^{2}-9}=\frac{x+3}{x-3} \cdot \frac{4 x}{(x+3)(x-3)}=\frac{4 x}{(x-3)^{2}}$$

Step-by-Step Solution

Answer

The correct equation should be: \(\frac{x+3}{x-3} \div \frac{4x}{x^{2}-9} = \frac{x+3}{x-3} \cdot \frac{(x+3)(x-3)}{4x}\) after factoring and simplification.

Step 1: Identify the Error

Take note of the equation and focus on the factoring error in the second fraction's denominator. Just to clarify, \(x^{2}-9\) must be factored into \((x+3)(x-3)\). This factoring is correct.

Step 2: Evaluate the Error

When dividing fractions, remember that division basically means to multiply by the reciprocal of the divisor. So, when dividing \(\frac{x+3}{x-3}\) by \(\frac{4x}{x^{2}-9}\), it should also be possible to express it as \(\frac{x+3}{x-3}\) times \(\frac{x^{2}-9}{4x}\). However, the given equation seems to suggest that the division of the two fractions results in a multiplication with a non-reciprocal fraction, which is incorrect.

Step 3: Correct the Error

Correct the error. When dividing \(\frac{x+3}{x-3}\) by \(\frac{4x}{x^{2}-9}\) it should be written as \(\frac{x+3}{x-3} \cdot \frac{x^{2}-9}{4x}\), factoring \(x^{2}-9\) gives \(\frac{x+3}{x-3} \cdot \frac{(x+3)(x-3)}{4x}\). Factor appropriately and simplify.

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill, particularly when simplifying rational expressions. A quadratic expression like \(x^2 - 9\) is often encountered in mathematics, and recognizing it as a difference of squares can greatly simplify your work.

Here's a quick recap on how it works:

• A difference of squares can be identified when you have an expression of the form \(a^2 - b^2\).
• This expression can be factored into \((a - b)(a + b)\).
• In our case, \(x^2 - 9\) is the same as \(x^2 - 3^2\), leading to the factors \((x - 3)(x + 3)\).

By factoring quadratics correctly, you lay the groundwork for simplifying more complex expressions. Always keep an eye out for the pattern of a difference of squares, as it is a powerful tool in factorization.

Division of Fractions

Understanding the division of fractions is crucial, especially when working with algebraic expressions. To divide one fraction by another, you simply multiply by the reciprocal of the divisor.

Here's how it works:

• Find the reciprocal of the second fraction. The reciprocal of \(\frac{4x}{x^2 - 9}\) is \(\frac{x^2 - 9}{4x}\).
• Change the division sign to a multiplication sign.
• Multiply the first fraction by this reciprocal. So now \(\frac{x+3}{x-3} \cdot \frac{x^2 - 9}{4x}\).

It is essential to always apply this rule when you encounter the division of fractions. Not using the reciprocal correctly is a common mistake, leading to incorrect simplifications.

Multiplication by Reciprocal

Multiplication by the reciprocal is not just a procedure for dividing fractions, it's a fundamental arithmetic rule that we use when dividing rational expressions as well.

This method allows:

• Fraction division to transform into a more straightforward multiplication problem.
• In our example, instead of dividing \(\frac{x + 3}{x - 3}\) by \(\frac{4x}{(x + 3)(x - 3)}\), you multiply by \(\frac{(x + 3)(x - 3)}{4x}\).
• This step is critical for correctly simplifying the expression, as skipping or misapplying it means the whole expression becomes incorrect.

By multiplying by the reciprocal, you align your expression into an equivalent form that is often simpler and easier to handle. Always remember, the reciprocal essentially "flips" the fraction around, turning division into multiplication, which is often easier to manage.